a preview of the Riemann mapping theorem
/ REE-mahn /
We have collected a toolbox of explicit conformal maps — Mobius maps, exp and log, Joukowski, Schwarz-Christoffel. A grander question now looms: how many regions can we hope to map conformally onto a disk? Could there be some strangely shaped region, all curves and inlets, that no conformal map can reach? The Riemann mapping theorem gives a breathtaking answer: essentially every reasonable region can be mapped onto the disk. It is the central existence theorem of the whole subject, and this entry is a preview — the full proof, using normal families, comes later.
Here is the precise statement. Let D be any simply connected open region in the plane that is not the entire plane (so it is a 'proper' subregion, and 'simply connected' means it has no holes — any loop inside can be shrunk to a point). Then there exists a conformal bijection f mapping D onto the open unit disk. Moreover, if you also specify that some chosen interior point of D goes to the center 0 and that f' is positive there, the map is unique. So from the point of view of conformal geometry, all such regions — a square, an amoeba-shaped blob, the inside of a wiggly Jordan curve, an infinite strip — are interchangeable with the disk and hence with each other. There is essentially only ONE simply connected region up to conformal equivalence (plus the two exceptions noted below).
Two honest caveats are essential, and both are famous. First, the theorem is non-constructive: it guarantees the map exists but does not produce a formula. Finding the explicit map for a given region can be hard or impossible in closed form — which is exactly why the explicit-map toolbox (Schwarz-Christoffel and friends) still matters. Second, the hypotheses cannot be dropped. 'Not the whole plane' is required because, by Liouville's theorem, any holomorphic map from the entire plane into the bounded disk must be constant, so the plane itself cannot be mapped onto the disk. And 'simply connected' is required because an annulus (a region with a hole) is genuinely NOT conformally a disk — you cannot fill in a hole with an angle-preserving map. The reward for accepting these conditions is enormous: it reduces the study of boundary-value problems on arbitrary simply connected regions to the single, fully understood case of the disk.
The interior of a square and the open unit disk look nothing alike, yet the Riemann mapping theorem guarantees a conformal bijection between them. (Building it explicitly needs the Schwarz-Christoffel formula, and the answer involves elliptic integrals — a vivid illustration that 'exists' and 'easy to write down' are very different things.)
Any proper simply connected region is conformally a disk — even when no elementary formula exists.
The two exclusions are not technicalities: the whole plane is excluded by Liouville's theorem, and a region with a hole (not simply connected) is excluded because holes are a conformal invariant. Simple connectivity is the precise dividing line.